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Properties

Label 2-91-7.2-c1-0-4
Degree $2$
Conductor $91$
Sign $0.936 + 0.349i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.10 + 1.91i)2-s + (−1.23 − 2.14i)3-s + (−1.44 − 2.51i)4-s + (1.06 − 1.83i)5-s + 5.47·6-s + (2.63 − 0.272i)7-s + 1.98·8-s + (−1.56 + 2.70i)9-s + (2.34 + 4.06i)10-s + (−2.39 − 4.14i)11-s + (−3.58 + 6.21i)12-s + 13-s + (−2.39 + 5.34i)14-s − 5.25·15-s + (0.697 − 1.20i)16-s + (1.88 + 3.27i)17-s + ⋯
L(s)  = 1  + (−0.782 + 1.35i)2-s + (−0.714 − 1.23i)3-s + (−0.724 − 1.25i)4-s + (0.474 − 0.822i)5-s + 2.23·6-s + (0.994 − 0.102i)7-s + 0.703·8-s + (−0.520 + 0.901i)9-s + (0.742 + 1.28i)10-s + (−0.721 − 1.25i)11-s + (−1.03 + 1.79i)12-s + 0.277·13-s + (−0.638 + 1.42i)14-s − 1.35·15-s + (0.174 − 0.301i)16-s + (0.458 + 0.793i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.936 + 0.349i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.936 + 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.577494 - 0.104217i\)
\(L(\frac12)\) \(\approx\) \(0.577494 - 0.104217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.63 + 0.272i)T \)
13 \( 1 - T \)
good2 \( 1 + (1.10 - 1.91i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.23 + 2.14i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.06 + 1.83i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.39 + 4.14i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.88 - 3.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.78 + 3.08i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.23 - 3.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.90T + 29T^{2} \)
31 \( 1 + (-1.88 - 3.26i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.81 - 4.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 3.40T + 43T^{2} \)
47 \( 1 + (-3.55 + 6.15i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.19 - 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.39 + 4.14i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.60 - 2.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.44 - 2.51i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.53T + 71T^{2} \)
73 \( 1 + (3.85 + 6.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.58 + 4.48i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 + (1.83 - 3.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.40T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.95499690662616336134306014104, −13.22899493257819188258363284456, −11.97565363803679616908655608875, −10.80061309196485148879829559611, −9.072974065715665633806741505527, −8.124011054761666687748373694087, −7.36181998675726873674072494321, −5.91328389419399910801102106711, −5.40016663764971863379182553645, −1.12783726141663635359438766510, 2.36060517314558161997451118530, 4.18767110669254397647783736591, 5.59642784637144022373725924927, 7.71290583146263995407545258438, 9.347926428421646456610868284682, 10.13881340757533615052462242536, 10.72464524775351696525340410892, 11.50884583225051438832565116121, 12.53078438802478121022677937612, 14.27337935298996128993016011834

Graph of the $Z$-function along the critical line